We will use a Cut Threshold procedure from our recent work [1]}. Recall that a Cut Threshold is defined as follows:

Let \(\lambda (s, t)\)  denote the value of the max-flow between two vertices \(s\)  and \(t\) ; we call \(\lambda (s, t)\)  the connectivity between \(s\)  and \(t\) .
Then, the Cut Threshold for vertex \(s\)  and threshold \(\phi \ge 0\) , denoted \({\rm ct}(s, \phi )\) , is the set of all vertices \(t\in V\setminus \lbrace s\rbrace \)  such that \(\lambda (s, t) \le \phi \) . 
In recent work, we gave an algorithm for finding a Cut Threshold  [1]} based on our isolating cuts framework [3]}:
[Li and Panigrahi [1]}]
Let \(G = (V, E)\)  be an undirected graph containing \(m\)  edges and \(n\)  vertices.
For any given vertex \(s\in V\)  and threshold \(\phi \ge 0\) , there is a randomized Monte Carlo algorithm for finding the Cut Threshold \({\rm ct}(s, \phi )\)  in \(\tilde{O}(m) + \operatorname{polylog}(n)\cdot F(m, n)\)  time, where \(F(m, n)\)  is the running time of any max-flow algorithm on undirected graphs containing \(m\)  edges and \(n\)  vertices.
